Computing the expectation of the Azéma-Yor stopping times

A NNALES DE L ’I. H. P., SECTION B

J. L. P EDERSEN

G. P ^EŠKIR

Computing the expectation of the Azéma- Yor stopping times

Annales de l’I. H. P., section B, tome 34, n

^o

2 (1998), p. 265-276

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265

Computing ^the expectation ^of

the Azéma-Yor stopping ^times

J. L. PEDERSEN

Institute of Mathematics, University ^{of Aarhus} Ny Munkegade, ^{8000 Aarhus, Denmark.}

E-mail: jesperl@mi.aau.dk

G. PE0160KIR

Institute of Mathematics, University ^{of Aarhus} Ny Munkegade, ^{8000 Aarhus, Denmark.}

(Department ^of Mathematics, University ^of Zagreb, Bijenicka 30, 10000 Zagreb, Croatia)

E-mail: goran@mi.aau.dk

Vol. 34, ^n° 2, 1998, p. 276 Probabilités et Statistiques

ABSTRACT. - Given the maximum process

(St)

⁼⁼

Xr)

associated with ^a diffusion

((Xt),P,),

^{and a} continuous

function g satisfying g ( s )

^{s, we show how to}

^compute

^the

expectation

^{of the}

Azéma- Yor

stopping

^time

as a function of x. The method of

proof

is based upon

verifying

^{that the}

expectation

^{solves a} differential

equation

^{with two}

boundary

conditions.

The third

’missing’

condition is formulated in the form of a

minimality principle

^{which states that the}

expectation

is the minimal

non-negative

solution to this

system.

^{It enables us to} express this solution in ^a closed form. The result is

applied

^{in the case when}

(Xt)

^{is a} Bessel process and g is a linear function. ©

Elsevier,

^Paris

Key ^{words and} phrases: Diffusion, the Azéma-Yor stopping time, ^the minimality principle,

scale function, infinitesimal operator, ^{the exit} time, ^normal reflection, Ito-Tanaka’s formula, Wiener process, Bessel process.

AMS 1980 subject classifications. Primary 60G60, ^60J60. Secondary 60G44, ^60J25.

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

34/98/02/@ Elsevier, Paris

RESUME. -

Etant

donnés une diffusion

((Xt),P,),

^{son maximum}

(St)

⁼

~r).

^{et une fonction}

continue g

^vérifiant

g(s)

^{5, nous}

montrons comment calculer

explicitement 1’ esperance

^du

temps

^{d’ arret} d’Azema-Yor

en tant que fonction de ^x. La méthode de demonstration utilise le fait que celle-ci est solution d’ une

equation

différentielle ^avec deux conditions ^{« au} bord ». Une troisième condition

sous-jacente

^{est formulée en terme d’un}

principe

^de

minimalité, lequel

énonce que ^cette

esperance

^est la solution minimale non

negative

^du

système.

^Ceci

permet d’ expliciter

^{cette solution}

comme une forme fermee. Nous

appliquons

^{ce résultat au cas ou}

(Xt )

^est

un processus de Bessel est g ^une fonction linéaire. (c)

Elsevier,

^Paris

1. FORMULATION OF THE PROBLEM

Let be ^a

non-negative

^{canonical diffusion with the}

infinitesimal

operator

^on

given by

where

and f-L

^are continuous functions on

(0, oc)

^and

^a2

^is furthermore

strictly positive (see [6]).

Assume there exists a standard Wiener process

( Bt )

such that for every x &#x3E; 0

The main purpose of this paper is ^to

compute

^the

expectation

^{of the}

Azéma-Yor

stopping

^time

(see [ 1 ] ).

^More

precisely,

for any continuous

function g

^on

satisfying

⁰

g( x)

^{x for x} &#x3E;

0,

the Azema-Yor

stopping

^time is defined as follows

where

(St)

is the maximum process associated with

(Xt)

started at s &#x3E; 0. The main aim of this paper is ^to present ^{a method for}

computing

the function

for 0 x s. Here the

expectation

is taken with

respect

^{to the}

probability

measure

P~

^:= under which the process

(Xt)

starts at x and the process

(St)

starts at s.

The motivation to

compute

^the

expectation

^{of such}

stopping

^{times comes}

from ^some

optimal stopping problems (see [2-4]

^and

[7]).

^{In these}

problems

it is of interest to know the

expected waiting

time for the

optimal stopping strategy

which is of the form T9 for ^some g. In view of this

application

we have assumed that the diffusion

(Xt)

^{and the function s ~}

g(s)

^are

non-negative,

but it will be clear from our considerations below that the results obtained are

generally

^valid.

The method of

proof

relies upon

showing

^{that the}

expectation

^{of the}

stopping

time solves a differential

equation

^{with two}

boundary

conditions.

The third

’missing’

condition is formulated in the form of a

minimality principle

^{which states that the}

expectation

^{is the minimal}

non-negative

solution to this system

(see Fig.

¹

^below).

^{It enables us to}

pick

up the

expectation

among all

possible

candidates in a

unique

way. The

minimality principle

is the main

novelty

^{in this}

approach (compare

^with

[2], [3]

^and

[4]).

In Section 2 the

minimality principle

^is

formulated,

and in Section 3 the existence and

uniqueness

of the minimal solution is

proved.

^{The main}

theorem is

proved

in Section

4,

^{and in} Section 5 an

application

^{of the}

theorem is

given.

2. THE MINIMALITY PRINCIPLE

In the first

part

of this section ^we shall observe that the function

~ t-~ solves a differential

equation

^{with two}

boundary

conditions.

In the

remaining part

of the section we will

present

^the

minimality principle

as the

’missing’ condition,

which will enable ^us to select the

expectation

of the

stopping

^{time in a}

unique

^way.

In the

sequel

^{we need the}

following

definitions and results. The scale function is for x &#x3E; 0

given by

Vol. 34, n° 2-1998.

where

We define as usual the first exit time from ^an interval

by

for 0 a

b,

^{and the}

following

formulas for 0 a x bare well-known

Let g

^{be a} continuous function

satisfying

⁰

g(x)

^{x for x} &#x3E; 0 such that ⁼ is finite for all 0 ~ s. We will now state the first result. Whenever s &#x3E; 0 is

given

^and

fixed,

the function x -

m( x, s )

solves the differential

equation

with the

following

^two

boundary

^conditions

A first

step

in the direction of

verifying

^that

(s, s)

^{~ satisfies}

the system above is contained in the

following

^result.

LEMMA 2.1. - The

function

^{s ~}

m(s, s)

^{- is}

^{C 1}

^and

satisfies

the

equation

Proof. -

^The

proof

^is

essentially

contained in Lemma 1 and Lemma 2 in

[3]. Alternatively,

^{to obtain a better}

feeling why (2.6) holds,

^{as well}

to derive it in another _way, ^one could use

(2.7)

^{below with}

(2.1 )+(2.2)

above to

verify

^that

equals

^the

right-hand

^{side in}

(2.6),

thus

showing

^that

(2.5)

^is

equivalent

^to

^(2.6),

and then follow the second

part

^{of the}

proof

of Theorem 4.1 below..

Let

g( s) x s be given

and fixed. It is

immediately

^{seen that}

and

by applying strong

^Markov

property

^we get

From

(2.1)

^and

(2.2)

^{we see that}

G(~) -

⁼

s)

^and

H(x)

⁼

^E~

^{solve the}

following

well-known

systems respectively

Consequently, by (2.6)-(2.9)

^we

easily verify

^{that x -}

m(x, s)

^{solves the}

system (2.3)-(2.5).

Note

since Tg

may be viewed ^as the exit time

by

^{diffusion from}

an open set, the

equation (2.3)

is well-known and the condition

(2.4)

^is

evident. The condition

(2.5)

is less evident but is known to be satisfied in

a similar context

(see [6]

p.

118-119).

Unfortunately (x, s)

^~--~

m(x, s)

^{is not}

uniquely

determined

by (2.3)

and the two

boundary

conditions

(2.4)

^and

(2.5).

^{Thus we} need another condition to determine

(x, s)

^~

m( x, s) uniquely.

^We formulate the third

’missing’

condition in the form of ^a

minimality principle (see Fig.

¹

below):

The

expectation m(x, s )

is the minimal

non-negative

^{solution to the}

system (2.3)-(2.5).

3. EXISTENCE AND

UNIQUENESS

^{OF THE MINIMAL SOLUTION}

Since

s )

^{= 0 for 0 x}

g ( s )

^we

only

^{need to consider m on}

9 ( s) x ~

^s.

Throughout

^we shall consider the

system

Vol. 34, n° 2-1998.

Motivated

by

^the

minimality principle,

ⁱⁿ this section we shall prove the existence

(and uniqueness)

^{of a minimal}

non-negative

^{solution to}

(3.1 ).

^Let

us introduce the

following

^notation:

where D ⁼

~(~~, 6’) ~(~) ~ ~’

^{6’, s} &#x3E;

0~

and D° is the interior of D. The function m

belongs

^to

C2u (D°) ni(x, s)

^is

^(7~

^{and s -}

s)

is

C’

on D° .

The main result of this section may be ^now formulated as follows. If Ji4 is

non-empty

^{then it contains a minimal}

element,

^i.e.

where the infimum is taken

pointwise.

Combined with the results in Section 2 this will be deduced in the

proof

of Theorem 4.1 I below

by using

^It6

calculus. The

proof

^we present here is based upon the

uniqueness

^theorem

for the first and second-order differential

equations.

For this note that the

uniqueness

^theorem

implies

^{that if m1 and m2}

belong

^{to .I~} then either rnl &#x3E; rn2 ^or mi rn2 ^on D°. Let

(xo, so)

^{E D}

be

given

^{and be a} sequence of functions from M such that

so) 1 so) .

^{Due to} the remark

just mentioned,

the sequence is

decreasing,

and therefore the limit exists

everywhere,

^i.e.

for all

(~B s)

^{E D. If we can show that}

then

using

^the

uniqueness

theorem it follows that m ⁼

In order to prove

(3.3)

^we first show that x ~ (x,

8)

solves the differential

equation

ⁱⁿ

(3.1). By

the instantaneous

stopping condition,

^and the

uniqueness ^theorem,

^{can be written as}

where s ^~ is ^a

~‘1-function.

Since is a

decreasing

sequence of

functions,

^the sequence is also

decreasing,

^{and therefore it}

converges

pointwise

^{to a function}

^A,

^{i. e.}

for ^- Hence ~~ ^~

.5)

^{solves the} differential

equation

ⁱⁿ

(3.1).

Annales de l’Institut Henri Poincaré - Probabilités

Obviously ~

^- satisfies the first

boundary

^condition

(instantaneous stopping)

since each j- t2014~

s)

satisfies this condition.

Finally,

^to

verify

the second

boundary

^condition

(normal reflection),

^note

that

straightforward computations

^{based on} the normal reflection condition and

(2.1 )+(2.2)

show that s -

An (s)

solves the

following

differential

equation

or

equivalently

Applying

^{the monotone} convergence

theorem,

^{we find that s -}

A(s)

^also

solves the differential

equation (3.4),

^{and we can} conclude that m E

4. THE EXPECTATION OF THE

AZÉMA-YOR

STOPPING TIMES The main result of the paper is contained in the

following

^theorem.

THEOREM 4.1. - Let

((Xt) , Pr)

^{be the}

non-negative diffusion defined

ⁱⁿ

( 1.1)

^{and let}

(St)

^{be the} maximum process associated with

(Xt ) defined

ⁱⁿ

(1.2).

^Let g be ^a

continuous function

^on

~0, oc) satisfying

⁰

g(x)

^~

for

x &#x3E;

0,

and let us

define

^the

stopping

^time

If

^M

from (3.2)

^is non-empty, ^{then is}

finite

^{and is}

given by

Vol. 34, n° 2-1998.

where

(x, s)

^1---+

s)

^is the minimal element in Jlit . The converse is also true, and ^we have the

following explicit formula

^{and a criterion}

for verifying

^{that M is} non-empty

for g(s)

^{~ s with = 0}

for

^{0 ~}

g(s),

^{which is valid}

in the usual sense

(if

^the

right-hand

^{side in}

(4.1 )

^is

finite,

^{then so is the}

left-hand ^side,

^{and vice}

versa).

Proof. -

^For

(xo, so)

^{E D}

given

^and

fixed,

consider the set

Choose bounded open ^sets

Gi

^C

G2

^{C ... such that}

Define the exit time of the two-dimensional diffusion

(Xt, St)

^from

^Gn by

Note that

Exo,so

^{00 and an}

^r

T~ ^{as n ~ oo.}

Let rn be any function in M. Note that ^m E

C2 ~’-

and

(St )

is of bounded variation so that Ito’s formula can be

applied (see

^{Remark 1 in}

[5]

p.

139).

In this _way we

get

Due to the normal reflection condition the last

integral

^is

identically

zero. Since the set of those u &#x3E; 0 for which ⁼

Su

^{is of}

Lebesgue

measure zero, and ⁼ -1 for

g(s)

^{x s, we can conclude}

that

Let be a localization for the local

martingale

Then

by

Fatou’s lemma and the

optional sampling

^{theorem we} get

Ex0,s0[m(X03C3n,S03C3n)] ~ lim inf Ex0,s0[m(X03C3n039BTk,S03C3n039BTk)] = m(x0,s0)

Thus ^we have the

inequality

for all n &#x3E;

1,

^and

by

^monotone convergence it follows

From this ^{we see} that is finite.

By

the results in Section 2 we

know that the function

(x, s)

^-

1

satisfies the

system (3.1 ),

^and

since m is

arbitrary,

^{hence we obtain}

This

completes

the first

part

^{of the}

proof.

To derive

(4.1 )

^{note that}

(2.6)

^{is a} first-order linear differential

equation

whose

general

solution is

easily

^{found to be}

given by

the formula

Vol. 34, n° 2-1998.

whenever the last

integral

^is

^finite,

^{where C is a real constant.}

Letting

s ^- oo we find that C = 0

corresponds

^to the minimal

non-negative

solution.

Combining

this with

(2.7)

^and

(2.1 )+(2.2),

^we obtain the

explicit

formula

(4.1 ).

^The

proof

of the theorem is

complete..

5. AN EXAMPLE

Let

( (Xt ) , P~ )

denote the Bessel process of dimension ^0152, where for

simplicity

^{we assume that 0152} &#x3E; 1 but 0152

#

^2.

(The

^{other cases of 0152}

could be treated

similarly.)

^Thus

(Xt)

^{is a}

non-negative

diffusion with the infinitesimal operator ^on

(0, oo) given by

(For

^more information about Bessel processes ^see

[5].) Let g

^{be a linear}

function

given by

where 0 A 1. Denote the

stopping time g by

It is ^our aim in this section to

present

^a closed formula for the

expectation

of the

stopping

^{time More}

precisely,

denote the function

for 0 x s. Then our main task is to

compute explicitly

^{the function}

Instead of

using (4.1 ) directly,

^we shall rather make ^use of the

minimality principle

within the

system (3.1 ).

According

^to Theorem 4.1 we shall consider the

system

Annales de l’Institut Poincaré - Probabilités

Let As x s be

given

and fixed. The

general

^{solution to}

(5.2)

^is

given by

where s -

A ( s )

^{and s -}

B ( s )

^are unknown functions.

By (5.3)

^and

(5.4)

^{we find}

whenever

(2 1a)~/(°~~)

^A

^{1 ,}

^where

and C is an unknown constant.

In order to determine the constant C we shall use the

minimality principle.

It is

easily

verified that the minimal

non-negative

^solution

corresponds

^to

C == O. Thus

by (5.5)-(5.7)

with C = 0 we have the

following

^candidate

for

when As x s. Hence

by applying

Theorem 4.1 we obtain the

following

result. Observe that this

example

is also studied in

[2]

^and

[3].

PROPOSITION 5.1. - Let

((Xt), Px)

^{be a Bessel} process

of

^{dimension a}

started at x &#x3E; 0 under

P~ ,

^{where a} &#x3E; 1 but c~

~

^{2. Then}

for

^the

stopping

time _Ta

defined

ⁱⁿ

(5.1 )

^{we have}

Vol. 34, n° 2-1998.

Fig. ^{1. - A} computer drawing of solutions of the differential equation (2.6) ^{in the case when}

a ⁼ 4 and A ⁼

4/5.

The bold line is the minimal non-negative ^solution (which ^{never hits} zero). By ^the minimality principle proved above, this solution equals ^ma (8,

s)

⁼

]

for all s &#x3E; 0.

REFERENCES

[1] J. AZÉMA and M. YOR, Une solution simple ^au problème de Skorokhod. Sém. Prob. XIII.

Lecture Notes in Math. 784, Springer-Verlag ^Berlin Heidelberg, ^1979, pp. 90-115 and 625-633.

[2] L. E. DUBINS L. A. SHEPP and A. N. SHIRYAEV, Optimal stopping rules and maximal

inequalities for Bessel processes. Theory ^Probab. Appl., ^Vol. 38, 1993, pp. 226-261.

[3] S. E. GRAVERSEN and G. PE0160KIR, Optimal stopping and maximal inequalities for linear diffusions. Research Report ^No. 335, Dept. ^Theoret. Stat. Aarhus (18 pp), ^{1995. To}

appear in J. Theoret. Probab.

[4] S. E. GRAVERSEN and G. PE0160KIR, On Doob’s maximal inequality for Brownian motion.

Research Report ^No. 337, Dept. Theoret. Stat. Aarhus (13 pp), 1995. Stochastic Process.

Appl., ^Vol. 69, 1997, pp. 111-125.

[5] D. REVUZ and M. YOR, Continuous Martingales and Brownian Motion. Springer-Verlag.

[6] L. C. G. ROGERS and D. WILLIAMS, Diffusions, ^Markov Processes, and Martingales; ^Volume

2: Itô’s Calculus. John Wiley &#x26; Sons, ^1987.

[7] L. A. SHEPP and A. N. SHIRYAEV, The Russian Option: ^Reduced Regret. ^Ann. Appl. ^Probab.

3, 1993, pp. 631-640.

(Manuscript ^received April 2I, 1997;

Revised 8, 1997.)

Annales de l’Institut Henri Poirzccire - Probabilités